Question

Simplify. $$\left[2 x^{2}\left(\frac{y^{3}}{w^{2}}\right)\right]^{2}$$

Short answer

\(4x^{4}y^{6}/w^{4}\)

Step-by-step solution

Apply the Power of a Product Rule

The Power of a Product Rule states that when you raise a product to a power, you can apply the exponent to each factor inside the product separately. So, the given expression \(\left[2 x^{2}\left(\frac{y^{3}}{w^{2}}\right)\right]^{2}\) can be simplified by raising each factor within the brackets to the power of 2.

Apply the Power to Each Factor

Raise each factor to the power of 2. This means: \(2^2\) to get 4, \(x^{2}\) raised to the power of 2 which becomes \(x^{4}\), and the fraction \(\left(\frac{y^{3}}{w^{2}}\right)^{2}\) which simplifies to \(\frac{y^{6}}{w^{4}}\).

Combine the Results

Combine the results of each factor raised to the power of 2. The earlier results 4, \(x^{4}\), and \(\frac{y^{6}}{w^{4}}\) can now be multiplied together to give the final simplified expression \(4x^{4}\left(\frac{y^{6}}{w^{4}}\right)\), which can also be written as \(\frac{4x^{4}y^{6}}{w^{4}}\).

Key concepts

Power of a Product Rule

When simplifying algebraic expressions, understanding the Power of a Product Rule is essential. It states that, when you raise a product (a set of numbers or variables multiplied together) to an exponent, the exponent applies to each factor individually. For example, if you have \( (ab)^n \), this is equivalent to \( a^n \cdot b^n \).

Let's apply this rule with numbers and variables to get a clear picture. If we take \( (2x)^3 \), this simplifies to \( 2^3 \cdot x^3 \) which equals \( 8x^3 \). The rule is powerful because it works for any number of factors inside the parentheses and with any exponent, greatly simplifying the process of exponentiation and making complex expressions more manageable.

Exponents

Exponents represent how many times a number or variable is multiplied by itself. The base is the number or variable being multiplied, and the exponent, positioned as a superscript, tells us the number of times the multiplication occurs.

For instance, \( x^5 \) means you have \( x \cdot x \cdot x \cdot x \cdot x \)—five times. It's important to note that exponents apply only to the base directly beside them, unless parentheses are involved, which then extend the range of the exponent. Exponents can make calculations faster and expressions much shorter.

Algebraic Simplification

The process of algebraic simplification is about making expressions as elementary as possible. This doesn't change the value of the expression; rather, it streamlines it for easier interpretation and calculation. The process involves a combination of applying exponent rules, like the Power of a Product Rule, combining like terms, and reducing fractions when possible.

For example, converting a complex fraction to a simpler form or expanding products to sum or simplify expressions by canceling out common factors are part of algebraic simplification. Mastering this requires practice and a solid understanding of the fundamental principles of algebra.

Math Problem Solving

Effective math problem-solving strategies are integral to successfully simplifying algebraic expressions. It often begins with understanding the given problem, identifying which rules and concepts apply, systematically applying those rules, and then checking the results for accuracy.

The ability to deconstruct complex problems into simpler components is critical. In the example above, recognizing that the Power of a Product Rule could be applied made the process straightforward. Additionally, cross-checking each step helps catch errors early and ensures the final solution is correct. Math problem solving is not just about finding an answer, it's about developing a methodical approach that can be applied to a wide range of problems.